Tag Archives: algebra

Learning intentions are more than just statements to convey to students what the learning is composed of; they are a means for building positive relationships with students. (Hattie, 48 pag.)

It is what I didn’t notice. The bell rang. As always, I heard a chorus of “Thank you, Ms. Gough. Bye, Ms. Gough.” It was normal practice – and a much appreciated practice – for my students to say thank you and goodbye as they left for their next class.

I thought to myself “what a great class, everything went well, and they are so nice.” I busied myself straightening my desk, organizing paper, and mentally listing off the things I needed to do before my next class rolled in. Eat lunch was at the top of the list.

Then, I sensed it. I was not alone. It is what I didn’t notice. There she sat, so still, except for the river of tears falling out of her beautiful, sad, green eyes. The river ran off the desk and pooled on the floor. “What is wrong?” I asked as I sat down beside her.

As I gently placed my hand on her arm, her shoulders began to shake as she said “I f..f..f..failed!” Whoosh, another flood of tears.

Now, she had not failed from my point of view. Her test score, damp as her test was now, showed a grade of 92 – an A. And yet, she deeply felt a sense of failure. As we sat together and looked at her work, we discovered that there was one key essential learning – in fact, a prerequisite skill – that caused her to stubble.

Tears, still streaming down her face, she said “I don’t know where I’m going wrong. I don’t miss this in class, but on the test, I fall apart.”

The point is to get learners ready to learn the new content by giving their brains something to which to connect their new skill or understanding. (Hattie, 44 pag.)

So, of course, the stumbling block for this sweet child is a known pain point for learners who master procedures without conceptual understanding. Consistently, she expanded a squared binomial by “distributing” the exponent – a known pitfall. #petpeeve

When our learners do not know what to do, how do we respond? What actions can we take – will we take – to deepen learning, empower learners, and to make learning personal?

Kamb’s insight was that, in our lives, we tend to declare goals without intervening levels. We declare that we’re going to “learn to play the guitar.” We take a lesson or two, buy a cheap guitar, futz around with simple chords for a few weeks. Then life gets busy, and seven years later, we find the guitar in the attic and think, I should take up the guitar again. There are no levels. Kamb had always loved Irish music and had fantasized about learning to play the fiddle. So he co-opted gaming strategy and figured out a way to “level up” toward his goal:

Level 1: Commit to one violin lesson per week, and practice 15 minutes per day for six months.

Level 2: Relearn how to read sheet music and complete Celtic Fiddle Tunes by Craig Duncan.

Level 3: Learn to play “Concerning Hobbits” from The Fellowship of the Ring on the violin.

Level 4: Sit and play the fiddle for 30 minutes with other musicians.

Level 5: Learn to play “Promontory” from The Last of the Mohicans on the violin.

BOSS BATTLE: Sit and play the fiddle for 30 minutes in a pub in Ireland.

Isn’t that ingenious? He’s taken an ambiguous goal—learning to play the fiddle—and defined an appealing destination: playing in an Irish pub. Better yet, he invented five milestones en route to the destination, each worthy of celebration. Note that, as with a game, if he stopped the quest after Level 3, he’d still have several moments of pride to remember. (Heath, 163-164 pgs.)

What if I’d made my thinking visible?

What if I’d connected this learning to how 3rd graders are taught multiplication of two digit numbers by decomposing into tens and ones. What if I’d connected this learning to how 3rd graders are also taught to draw area models to visualize the distributive property?

What if I’d shared my thinking and intentionally connected prior learning in levels?

By using Kamb’s level-up strategy, we multiply the number of motivating milestones we encounter en route to a goal. That’s a forward-looking strategy: We’re anticipating moments of pride ahead. But the opposite is also possible: to surface those milestones you’ve already met but might not have noticed. (Heath, 165 pag.)

How might we help our learners level up, experience success at several motivating milestones, and notice successes that might otherwise go unnoticed?

By multiplying milestones, we transform a long, amorphous race into one with many intermediate “finish lines.” As we push through each one, we experience a burst of pride as well as a jolt of energy to charge toward the next one. (Heath, 176 pag.)

Taken together, these practices make learning visible to students who understand they are under the guidance of a caring and knowledgeable teacher who is invested in their success. (Hattie, 48 pag.)

Does a student know that they are confused and can they express that to their teacher? We need formative assessment and self-assessment to go hand-in-hand.

I agree that formative self-assessment is the key. Often, I think students don’t take the time to assess if they understand or are confused. I think that it is routine and “easy” in class. The student is practicing just like they’ve been coached in real time. When they get home, do they “practice like they play” or do they just get through the assignment? I think that is where deep practice comes into play. If they practice without assessing (checking for success) will they promote their confusion? I tell my students that it is like practicing shooting free throws with your feet perpendicular to each other. Terrible form does not promote success. Zero practice is better than incorrect practice.

With that being said, I think that teachers must have realistic expectations about time and quality of assignments. If we expect students to engage in deep practice (to embrace the struggle) then we have to shorten our assignments to accommodate the additional time it will take to engage in the struggle. We now ask students to complete anywhere from 1/3 to 1/2 as many problems as in the past with the understanding that these problems will be attempted using the method of deep practice.

Our version of deep practice homework:
“We have significantly shortened this assignment from years past in order to allow you time to work these questions correctly. We want you do work with deep practice.

Please work each problem slowly and accurately.

Check the answer to the question immediately.

If correct, go to the next problem.

If not correct, mark through your work – don’t erase – leave evidence of your effort and thinking.

Try again.

If you make three attempts and can not get the correct answer, go on to the next problem. “

I also think that the formative assessments with “leveling” encourage the willingness to struggle. How many times has a student responded to you “I don’t get it”? Perhaps it is not a lack of effort. Perhaps it is a lack of connected vocabulary. It is not only that they don’t know how, is it that they don’t know what it is called either. It is hard to struggle through when you lack vocabulary, skill, and efficacy all at the same time. How might we help our learners attend to precision, to communicate in the language of our disciplines?

Now is the time to guide our young learners to develop voice, confidence (and trust), and a safe place to struggle.

Like this:

How can we promote success-oriented behaviors to foster learning and self-efficacy?

I dare you to read the following journal entries but replace the word math with assessment or whatever you are struggling to learn right now. Out of the mouth of babes…

I’ve been rereading journal entries from August to reflect on the growth of children I coach to learn algebra. The point of this particular journal entry was to help assess disposition.

Can we effect their growth in algebra AND their growth as learners? Can changing our assessment practices and our approach to learning help them learn to embrace the struggle, to see that a “failure” is an opportunity to learn? Does success breed success? Does success change your confidence, efficacy, and disposition?

How can we help failure-avoidant students grow to become success-oriented learners? Are most learners both success-oriented and failure-avoidant with a strong preference for one or the other?

Wait… I choose to revise my question. How can we promote success-oriented behaviors to foster learning and self-efficacy?

What do you think?
Is QB success oriented, failure avoidant, or both?

”The reason why I chose a picture of a person repelling or climbing a mountain is because math is a mountain for me. A mountain is an object that you cannot go through or around. The only way to get to the top of the mountain is by climbing. Math for me is a mountain. I can only climb my way to the top. There will be slips and falls along the way but, that is the only way to get to the top of the mountain. Every step I take teaches me something about that mountain. When you climb to the top of the mountain you can look back and say all those little slips and falls taught me something about that mountain, but now I can see all those tiny steps added up.”

“Every step I take teaches me something about that mountain. When you climb to the top of the mountain you can look back and say all those little slips and falls taught me something about that mountain, but now I can see all those tiny steps added up.”

I love this child; he spends many hours with me learning and improving. We have two classes together, and he chooses to work with me after school several days each week. When I read his journal on the first day of class, I put him in the success-oriented category. As I have worked with him this semester, I have seen him on a rollercoaster ride, struggling to not lapse into failure-avoidant behaviors. I believe it is my job is to be his cheer-master, his coach, and his support. I want to coach him to find his strenghts and successes.

The same day, CL wrote:

“I think this picture best describes my experiences in math for a lot of reasons. If you look at the girl’s face, it seems like she doesn’t know what she is doing. But if you look at her body, she seems to be doing the right thing. This is like me in math in a way. A lot of times I am doing the right steps, but I still think I am wrong. Like the girl in the photo, I don’t believe I am doing the right steps (or moves in her case). My feelings toward math are basic. I don’t love math, but I don’t hate it. Math also doesn’t come naturally to me. I have to work hard at something until I really understand it. I am more interested in math that we use every day than just random lessons. I also like to know the why in things. Like “Why do we use this trick?”. The why and how are keys words for me in learning math. I think my job in math is to learn new things, listed to the students and other students and the teachers, and to help others learn. I believe math is very helpful in everyday situations. I also believe math is hard, but if you work hard enough you will understand it. I want to learn from my mistakes in math. I also need different techniques to learn from if one doesn’t work. Lastly, my goal this year in math is to maintain a high grade by fully understanding the material.”

How often do we make curricular decisions based on what we think we see? Are we looking at the face or the body? How often do we assume that our students are learning? Do we check for evidence of learning – not grade – really check for proof? When we see the body doing the right things, do we ignore the face? Do we check for confidence? I fear that we may promote failure-avoidant behaviors if we are not careful.

CL wrote:

“If you look at the girl’s face, it seems like she doesn’t know what she is doing. But if you look at her body, she seems to be doing the right thing.”

How do we give our learners enough feedback so that they know that they are doing the right work? How do we build up their confidence so that they will either feel successful or know that it is safe (and encouraged) to ask questions to learn and grow? How do we reward effort and willingness to struggle to learn without giving students a false impression of their achievement?

CL:

“I want to learn from my mistakes in math. I also need different techniques to learn from if one doesn’t work.”

Me too! If we don’t assess learning and offer feedback in the midst of the experience, how will we know if we are promoting learning for all? How will we know if some (or all) need a different approach? Again, we must be careful to promote success-oriented behaviors.

I also think that my team and I spend a fair amount of time in CL’s shoes.

“A lot of times I am doing the right steps, but I still think I am wrong. Like the girl in the photo, I don’t believe I am doing the right steps (or moves in her case).”

Am I doing the right things for my students? My assessment plan is so different from what they will probably experience next year. When I listen to others who are uncomfortable with this “radical” change, I question if I’m doing the right steps. From what I read and study, I believe that I am doing the right things to help them learn and grow.

CL’s words where I have replaced math with assessment:

I don’t love assessment, but I don’t hate it. Assessment also doesn’t come naturally to me. I have to work hard at something until I really understand it.

My team experiments with me. Are we failure-avoidant teachers or success-oriented learners? We collect data and ask questions; We refine our hypothesis and try again. We are learning by doing; we are making assessment and grading decisions based on what the data indicates. Are we confident about our assessment work 100% of the time? No…Does it cause us to ask questions, think deeply, risk, learn? Yes…

Video from NPR. (Watch the video, seriously; it’s only 2:34 and well worth it!)

So…Which is most appealing to you? Which draws you in and generates questions? I think we need all of the above, the hands-on investigation of the data with technology, the infographic that gives perspective, and the video that offers an alternate way to visualize and think about this population growth.

How are we “leveling up” concerning visualization? Have our learners been introduced to infographics? Better yet, have our learners produced infographics to communicate data creatively? How are we using video to engage our learners? Have our learners produced video to communicate data, learning, and growth? Are we teaching (and learning) Information Age skills if we are not expecting multiple representations of ideas from our learners?

So… with lots of technology at our fingertips, if a picture is worth 1000 words, what is video worth?…and…what if we only communicate with text? What learning is lost when/if we only offer one representation of what we want others to learn?

How do students reflect on their work? What opportunities are offered to help students carry the essential learnings from first semester through second semester and/or into the next level of learning?

I’m interested and curious about different strategies and methods used to help learners process and reflect on their exam experience and the accumulation of what they know. Since each learner will have different bright spots and strengths, what strategies are used to differentiate for intervention and enrichment?

We aim to get “in the weeds” about reflection and intervention. We want every child to reflect on what they could demonstrate well and where they need additional help. We do not want them to move to the next year with any doubt or weakness if we can help now. But, how do we know who needs help? We collect data, and we let our learners gather data. We need to be informed; they need to be informed. We are a team working toward the goal of mastery or proficiency for all learners.

Our process:

Return the exam to the learner on the first day back.

Have each learner complete the exam analysis and reflection form (shown below) to identify strengths and areas of need.

At T³, Sam and I also facilitated a 90-minute session titled Ask, Don’t Tell: Listen to Learn and Assess. Here’s the program description and our simple agenda.

Ask, Don’t Tell: Listen to Learn and Assess Can we merge diagnostic and formative assessment to lead learning? How will TI-Nspire™ CAS Handheld action-consequence documents combined with the TI-Nspire™ Navigator™ System allow us to leverage technology to focus on learning? What if we used the ideas of simplicity and restraint when developing and leading lessons? What can be learned if we question our way through an entire lesson? is it possible to allow students to steer the lesson through their questions? Will listening to student questions help us diagnose, assess and chart a course in real-time? Can we lead learning by following their thinking? Will you come to this session and plan to serve as a student, an observer, and a questioner?

Sam then introduced the Ask; Don’t Tell idea by modeling a lesson on the discriminant using the TI-Nspire Quadratic_Roots.tns file and the 3-12-3 protocol.

Want to explore the investigation? Here’s how: Clicking on the screenshot should enable you to download the TI-Nspire document and open it if you have the TI-Nspire software on your computer. Clicking on the Launch Player button should open a player file where you can interact with the document without having TI-Nspire software. (Be patient; it is a little slow to launch.)

12 minutes: Work with a partner to share questions, convert closed questions to open questions, and generate additional questions. Partners should identify their top 2-3 questions.

3 minutes: Use the TI-Nspire Navigator to collect each student’s top question.

Facilitate the class discussion of the lesson by responding to student questions from students as well as the teacher.

Following his “lesson,” Sam check for understanding using the leveled QuadInvestForm.tns formative assessment.

Again, great discussion from our participants. Sam received good feedback about his assessment. Participants shared strategies they have used to debrief student responses while using the Navigator. I thought it was great that Sam opened the discussion up by asking for ideas from the participants.

After experiencing a leveled assessment, I facilitated a discussion about the philosophy and strategies involved in using this type of formative assessment. The summary of this discussion was captured by Sarah Bauguss (@SBauguss).

I am grateful that Sarah took the time to tweet during the session. Often I don’t really know what I conveyed. Having this series of tweets offers me another level of feedback.

Our educational systems have been constructed entirely around the goal of providing the correct answer to a question provided by an instructor or handed out on a standardized exam. This system provides a form of valid comparison for the results of a group of students, and it provides a foundation of shared information amongst those who have followed a course of study. Unfortunately, the real world, particularly the real world of the coming century, does not and will not work this way. Our heroes are not defined by how well they answered canned questions or what they scored on their SATs precisely because these outcomes do not determine success in real-world situations. The real revolution in education and training, if it comes, will be overtly switching our priority from the skills of giving answers to the skills of finding new questions.

Questions are waypoints on the path of wisdom. Each question leads to one or more new questions or answers. Sometimes answers are dead ends; they don’t lead anywhere. Questions are never dead ends. Every question has the inherent potential to lead to a new level of discovery, understanding, or creation, levels that can range from the trivial to the sublime. (Lichtman, 35 pag.)

Grant told two powerful stories of leveraging learner questions to facilitate learning. He made the great point that if you teach from student questions, you know someone in the room is interested in what is being discussed.

Then it got seriously interesting for us. Grant facilitated an experience of questioning techniques while I drove a lesson (shown below) on the TI-Nspire.

Want to explore the investigation? Here’s how: Clicking on the screenshot should enable you to download the TI-Nspire document and open it if you have the TI-Nspire software on your computer. Clicking on the Launch Player button should open a player file where you can interact with the document without having TI-Nspire software. (Be patient; it is a little slow to launch.)

Grant reviewed the “big 6” types of questions and transitioned to another type of question – “What if”. Here’s what the exercise looked like when I finished following his directions. Remember, he could not see the linear investigation, and I could not see him.

Grant then signed off so that we could roll up our sleeve and get to work experiencing learning through the art of questioning. I opened the next section of this lesson by reading from Step 0: Preparation of The Falconer: What We Wish We Had Learned in School.

The excitement of learning, the compelling personal drive to take one more step on the path towards wisdom, comes when we try to solve a problem we want to solve, when we want to solve, when we see a challenge and say yes, I can meet it. Great teachers lead us just far enough down a path so we can challenge for ourselves. They provide us just enough insight so we can work toward a solution that makes us, makes me want to jump up and shout out the solution to the world, makes me want to step to the next higher level. Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome. (Lichtman, 20 pag.)

Wow! Worth repeating:

Great teachers lead us just far enough down a path so we can challenge for ourselves.

So, how do we do this? Sam accepted the challenge of modeling this type of facilitation of learning by leading a lesson. We wanted the participants to experience the investigation, question generation, and learning. Sam chose to use the EllipseInvest.tns file show below.

12 minutes: Work with a partner to share questions, convert closed questions to open questions, and generate additional questions. Partners should identify their top 2-3 questions.

3 minutes: Use the TI-Nspire Navigator to collect each student’s top question.

Facilitate a class discussion by responding to student questions encouraging responses from students as well as the teacher.

It was awesome! Sam knew that he was going to administer a formative assessment next. As his peer observer, I could see his effort and questioning to guide the discussion through the participant questions to the essential outcomes of the lesson. Another point from The Falconer that is worth repeating:

Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome.

Experientially, our participants could make their own determination of the value of this type of formative assessment. Continuing his questioning technique, Sam prompted the participants to identify why the questions were at the given level. Could they see a leveling up in the questions? What did it take to move from one level to another? The discussion was excellent, and Sam received strong feedback about his assessment design. Yay!

We were at about the 2-hour mark in our 4 hour workshop. I asked our participants if they could stand a 4-minute Ignite talk on assessment to set the stage for the next 2 hours of work and then we would take a break.

Leveled assessments provide the opportunity to bright spot the work of every learner. They come in the door saying I can do this, Ms. Gough; will you help me level up? Let’s take the challenge of highlighting what learners can do rather than what they cannot.

For the next hour, Sam and I watched, listened, and coached as participants worked to designed a leveled assessment on a topic of their choosing. We displayed an example through the projector as a point of reference. Our participants asked for the template of the table of specifications. All files linked on my previous post are .pdfs. The table of specifications as a Word doc is shared below.

How great that our participants asked for a usable resource! Learn and share!

I was so pleased with the engagement and the collaboration of our participants. There were so many good questions. I challenged our participants to share their ideas – in any form by dropping their files in my Dropbox. I’ve promised to zip all files shared in my Dropbox by Tuesday and share this on this post.